Unlike most modern public-key ciphers, whose security relates to some long-studied mathematical problem that is believed to be difficult to solve (e.g., the factoring or finding discrete logarithms of large integers), the security of most modern symmetric-key block ciphers do not relate to any widely-studied, hard-to-solve problems. Rather, block ciphers are generally designed in an ad hoc fashion to resist known cryptanalytic attacks. As a consequence, the design, analysis and implementation of reliably secure block ciphers is regarded as exceptionally difficult, and is often regarded as more of an art than a science.
The lack of a system and method for developing efficient block ciphers whose reliability and security can be understood or expressed analytically has resulted in the deployment of block ciphers whose security properties have been discovered to be considerably weaker than expected. The discovery of a new weakness in a block cipher undermines security of systems in which it is used, and results in inconvenience and economic loss if the discovered weakness is severe enough to warrant replacing the cipher.
A more serious problem arises if the cipher's user is unaware of a weakness that has been discovered by a third party. This weakness may be exploited without the knowledge of the user to undermine the security of the user's systems for an indeterminate amount of time. This can lead to the unauthorized modification of information (such as the dollar amounts of transfers specified by electronic funds transfer (EFT) messages) and/or the disclosure of confidential and sensitive information to unauthorized third parties (e.g., the disclosure of a trade secret.) Such security compromises can cause significant damage to the user and to third parties who rely upon the security of the cipher indirectly (e.g., account holders at a bank that uses EFT secured by the cipher.)
A block cipher with known security properties would eliminate much of the uncertainty surrounding its security. This would substantially reduce the risk of an unexpected weakness, allowing users to rely upon it with more confidence. A cipher with security properties known to be strong would reduce the risk of unauthorized modification and/or compromise of confidential and/or sensitive information. Michael R. Garey and David S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, 1979.
Known block ciphers tend to be conceptually complex. They are often characterized by "magic" constants (i.e., apparently arbitrary constants that have a poorly understood effect on the security of the cipher), irregular structures, and awkward bit-level operations that are inefficient and expensive to implement on computers and/or in telecommunications systems. It is virtually impossible to mathematically comprehend the justifications for many of the various parameters in a typical cipher. These features can lead a user to improperly implement the cipher in software and/or hardware. Improperly implementing even a single step in some ciphers can render them far less secure. For example, a single bit change to a DES S-box in the Digital Encryption Standard (DES) cipher usually renders the cipher less secure.
Cryptanalytic attacks against known block ciphers have often been successfully carried out using known mathematical techniques. Eli Biham and Adi Shamir, Differential Cryptanalysis of the Data Encryption Standard, Springer-Verlag, 1993. A block cipher that cannot be successfully attacked with known mathematical techniques would be more secure than many known block ciphers.
The goal of a block cipher is to provide a reversible transformation on blocks of bits. More precisely, block ciphers are reversible pseudo random permutations that map each of the 2.sup.n possible inputs to a unique n-bit output value. An ideal block cipher would be a completely random permutation, i.e., the only possible representation of the transformation would be a list that completely maps each possible input to an output, and vice versa. This is called a "random function." An example of such a random function for three bit blocks (each of which was selected at random) is as follows:
______________________________________ INPUT 000 001 010 011 100 101 110 111 OUTPUT 010 111 000 110 100 001 011 101 ______________________________________
A truly random function is said to be unconditionally secure because there is no correlation between any one subset of mappings and any other, i.e., the most compact representation of the function is simply a list of all of its inputs mapped to outputs. However, the size of such a random function grows exponentially with the number of bits on which the cipher operates. For a block size of 2.sup.n bits, a block cipher that operates on two halves of the block requires at least one random function correlating 2.sup.n inputs to 2.sup.n outputs, and generally more in order to provide meaningful security. This makes random functions impractical to implement for block ciphers.
This problem has been addressed by replacing the use of random functions with pseudo random functions (PRF) in a known block cipher developed by Feistel. H. Feistel, "Cryptography and Computer Privacy." Scientific American, Vol. 228, 1973. A PRF is not a truly random mapping of inputs to outputs, but rather a function that generates an apparently random mapping based upon a particular method. Luby and Rackoff showed that if the functions used in an at least four-round Feistel construction are themselves secure, then the resulting permutation is secure. M. Luby and C. Rackoff, "How to Construct Pseudo random Permutations from Pseudo random Functions." SIAM J. Comput., 17 (1988), 373-386. However, Luby and Rackoff provided no information on how to determine if an arbitrary PRF is in fact secure. Thus, a system and method that uses functions known to be secure in an at least four-round Feistel construction will produce permutations that are known to be secure. The Feistel primitive is shown symbolically as follows: EQU X=A.vertline.B /*cleartext*/ EQU A=A+f.sub.e EQU B=B+A EQU X =B.vertline.A /*ciphertext*/
X=A.vertline.B indicates that block of data X has two concatenated halves A and B (".vertline." indicates concatenation). Entry f.sub.e of PRF f is added in this embodiment bitwise modulo-2 (represented by the "+" operator) to A. The result is assigned to A, which is then added to B modulo-2, resulting in a new value for B. The positions of A and B are switched and concatenated to form a permuted X=B/A. The primitive can repeated again any number of times using a different PRF each time. Each instance that the primitive is invoked is called a "round." A symbolic representation of a four round Feistel construction is as follows: EQU X=L.vertline.R /*cleartext*/ EQU R=R+f.sub.1 (L) EQU L=L+f.sub.2 (R) EQU R=R+f.sub.3 (L) EQU L=L+f.sub.4 (R) EQU X=L.vertline.R /*ciphertext*/
Here, f.sub.1, f.sub.2, f.sub.3 and f.sub.4 are secret pseudo random functions. A four round, 2n-bit Feistel construction using four n-bit functions is shown in FIG. 1. In step 1, a 2n-bit block X is divided into a right half R and a left half L. In step 2, the n-bit left half L of the 2n-bit block is permuted with a PRF f.sub.1 and added to the n-bit right half R of the block. The result becomes the new right half R. In step 3, the permuted right half R is permuted with another PRF f.sub.2 and added to left half L. The result becomes the new left half L. In step 4, the L is permuted with another PRF f.sub.3 and added to R, the result of which becomes the new R. In step 5, R is permuted with PRF f.sub.4 and added to L, the result of which becomes the new L. In step 6, an enciphered block X=L.vertline.R is obtained.
In order to decipher a block enciphered with the Feistel primitive, the order of the steps of the primitive are reversed and carried out on the enciphered block. This is shown for the Feistel primitive as follows: EQU X=B.vertline.A /*ciphertext*/ EQU B=B+A EQU A=A+f.sub.e EQU X=A.vertline.B /*cleartext*/
In order to decipher blocks enciphered with multiple rounds, the rounds should be reversed in reverse order, the most recently used round first. In other words, if the primitive is applied in the sequence r1, r2, . . . , r5, the reverse primitive should be applied in the order r5, r4, . . . , r1 to decipher the block. This is shown for a four round Feistel construction cipher in FIG. 2, and as follows: EQU X=L.vertline.R /*ciphertext*/ EQU L=L+f.sub.4 (R) EQU R=R+f.sub.3 (L) EQU L=L+f.sub.2 (L) EQU R=R+f.sub.1 (L) EQU X=L.vertline.R /*cleartext*/
The Feistel construction cipher is advantageous because its security (using at least 3 rounds, more preferably at least 4 rounds, and most preferably at least 6 rounds) is closely related to the difficulty of solving the Numerical Matching with Target Sums (NMTS) problem, an NP-Complete problem which cannot be analytically solved using known mathematical techniques. In other words, the only known way to compromise a Feistel construction cipher of three or more rounds is by brute force (e.g., trying all possibilities.) The difficulty of succeeding in a brute force attack is related to the block size of the cipher, i.e., the larger the block size, the greater the security of the cipher is likely to be. The disadvantage of the Feistel construction is that it is impossible to practically implement for block sizes sufficiently large to render a secure cipher. This is due to the large computer-readable memory requirements imposed by a Feistel construction cipher for a secure block size.
As discussed above, the number of entries in a PRF needed to implement a Feistel primitive on a block of 2n bits is 2.sup.n. The number of bits of PRF needed is therefore n2n. For a construction using N rounds of the primitive on each block, Nn2n bits of PRF are required. It is important to keep the PRF confidential, because an unauthorized third party could use the PRF with the primitive reversed to decipher the block.
The basic 4-round Feistel construction is not suitable in practice for use as a practical block cipher because it requires 4n(2.sup.n) bits of secure (secret) memory to operate cryptographically on a block of length 2n. This limits the economical and practical application of the four round Feistel construction to 16 or 32 bit block sizes on known computers. For example, a four-round Feistel construction on a block of 64 bits requires around 2.sup.32 bits of storage memory (over 3 billion bits, or about 375 Mb), which exceeds the ready storage capacity (e.g., RAM) of many present day computers. For an even larger block cipher of 128 bits, 2.sup.64 bits (over 10.sup.19 bits) are required, roughly equal to the number of seconds in the age of the universe as it is presently understood.
A larger block size in a Feistel construction block cipher generally yields a more secure cipher because the PRFs of small block size ciphers can be easily deduced using known cryptanalytic techniques implemented on computers. This becomes more clear by considering a block size of three bits. Each input entry for the PRF has only eight possible output entries, as can be seen in the table above. All of the output possibilities may be easily tried for the inputs using a computer on a 3-bit block cipher, and the cleartext can be easily recovered. The present state of art mandates a minimum block size of 64 bits in order to produce a secure cipher, a block size which, as shown above, is impractical to implement using the four round Feistel construction.
It should also be recalled that Luby and Rackoff showed that an at least four-round Feistel construction is secure if the underlying functions are themselves secure, but no information was provided on determining the security of underlying PRFs. Hence, a practical Feistel construction using at least four rounds and whose underlying functions are known to be secure (e.g., whose underlying functions are random functions, rather than PRFs) would produce secure permutations. However, the constraints of computer readable memory in modern computers prevent the implementation of a four round Feistel construction using truly random, unconditionally secure functions.
In summary, a better block cipher would possess the advantage of having known security properties such as those disclosed by Luby and Rackoff for the Feistel construction, and could be efficiently and practically implemented on known computers.